Matrix Projective Synchronization Research Articles

Matrix projective synchronization and mechanical analysis of unified chaotic system

This article explores the mechanical analysis of a unified chaotic system and matrix projective synchronization (MPS). The sufficient conditions to achieve MPS of unified chaotic system have derived. The mechanics of unified chaotic system have been examined in contrast with Kolmogorov system, Euler equation, and Hamiltonian function. The Casimir energy function is also introduced to analyze the system dynamics. The unified chaotic system has been transformed into Kolmogorov type system and decomposed into four types of torques: inertial, internal, dissipation, and external torque. In order to view the mechanics behind the unified chaotic system, different particular cases have been discussed. Five scenarios are examined using combinations of various torques in order to identify the key elements in chaos creation and their physical significance. The bifurcation diagram, Lyapunov exponent and Lyapunov dimension validates the generation of chaos for different particular cases.

Matrix and Inverse Matrix Projective Synchronization of Chaotic and Hyperchaotic Systems with Uncertainties and External Disturbances

Matrix and Inverse Matrix Projective Synchronization of Chaotic and Hyperchaotic Systems with Uncertainties and External Disturbances

Function matrix projective synchronization for unknown and delayed fractional-order neural network

Function matrix projective synchronization for unknown and delayed fractional-order neural network

Function matrix projection synchronization for the multi-time delayed fractional order memristor-based neural networks with parameter uncertainty

Fractional Order Memristor-Based Neural Networks (FOMBNNs) has the strong sensitivity to initial values and shows more complex paths, so its Projective Synchronization (PS) and applications have been widely used in the fields of security communication. Our main work intends to extend the scaling factor of PS to a function matrix depending on time t and proposes a new synchronization type for the first time, i.e., Function Matrix Projective Synchronization (FMPS) for FOMBNNs, whose scaling factor is highly variable over time and difficult to predict. However, the FOMBNNs is a state dependent discontinuous system and it is easy to produce complex nonlinearity, which makes the study of the FMPS becomes a challenge. Therefore, our work will commit to solving this problem and realizing the FMPS for multi-time delayed FOMBNNs with parameter uncertainty. Firstly, the error functions and FMPS are defined, which can be degenerated to matrix PS, modified PS, PS, anti-synchronization and complete synchronization. Then, for the multi-time delayed FOMBNNs with parameter uncertainty, the active controller is designed and the sufficient condition for realizing the FMPS is proved by using a Lyapunov functional and some Lemmas of fractional calculus. Finally, the FMPS of four numerical examples are given and trajectories of their synchronization errors approach to 0, which illustrate the efficiency of the proposed synchronization analysis. This research will provide a general method for studying the FMPS of other dynamical systems.

Dual Function Matrix Projective Synchronization for Fractional-order Hyperchaotic Systems

Abstract It is well known that the variability and complexity of projection proportionality factors of Dual Projective Synchronization (DPS) can effectively enhance signal confidentiality. However, in most literatures, the proportionality factors are some simple fixed constants, which can't ensure high security of information. For two pairs of fractional-order hyperchaotic systems (FOHS), how to expand the projection proportionality factors to increase its complexity? Then, our work will propose a new synchronization type, i.e., Dual Function Matrix Projective Synchronization (DFMPS) and realize the DFMPS for FOHS for the first time. Firstly, based on the traditional DPS, we generalize the proportionality factors to a general function matrix depending on time t, present the error functions and define the DFMPS. Then, for FOHS, the active controller and synchronization condition are designed and proved. At the same time, when the system is affected by parameter disturbance, the proposed active controller can eliminate the influence of parameter disturbance to the system's DFMPS, which indicates that the proposed control strategy has strong robustness. Finally, the DFMPS of two pairs of fractional-order hyperchaotic Chen and Rabinovich systems are realized, and synchronizing analysis and system robustness analysis are verified by numerical simulation. Particularly, the DFMPS can be degenerated to dual anti-synchronization, dual complete synchronization, DPS, modified DPS and dual matrix projective synchronization. This work extends the synchronization types for FOHS and offers a useful method to explore DFMPS for other fractional-order systems.

Global matrix projective synchronization of delayed fractional-order neural networks

Global matrix projective synchronization of delayed fractional-order neural networks

Matrix projective synchronization between time delay chaotic systems with disturbances and nonlinearity

Matrix projective synchronization between time delay chaotic systems with disturbances and nonlinearity

Fixed-Time Adaptive Time-Varying Matrix Projective Synchronization of Time-Delayed Chaotic Systems with Different Dimensions

This paper deals with the fixed-time adaptive time-varying matrix projective synchronization (ATVMPS) of different dimensional chaotic systems (DDCSs) with time delays and unknown parameters. Firstly, to estimate the unknown parameters, adaptive parameter updated laws are designed. Secondly, to realize the fixed-time ATVMPS of the time-delayed DDCSs, an adaptive delay-unrelated controller is designed, where time delays of chaotic systems are known or unknown. Thirdly, some simple fixed-time ATVMPS criteria are deduced, and the rigorous proof is provided by employing the inequality technique and Lyapunov theory. Furthermore, the settling time of fixed-time synchronization (Fix-TS) is obtained, which depends only on controller parameters and system parameters and is independent of the system’s initial states. Finally, simulation examples are presented to validate the theoretical analysis.

Matrix projective synchronization for a class of discrete-time complex networks with commonality via controlling the crucial node

This article proposes two control strategies to realize matrix projective synchronization (MPS) for a class of discrete-time complex dynamical networks (DTCDNs). It is worth pointing out that our network model is composed of nodes with different state dimensions and its outer coupling configuration matrix can be asymmetric. In addition, since a small percentage of nodes that play more important role than others usually exist in a network, thus, one important node, which is named as crucial node in this paper, is taken into account in our network model. Besides, the commonality between each node and the crucial node is also precisely expressed. Further, based on the crucial node and the commonality and associated with the Lyapunov stability theory, two control strategies are addressed to realize the exponential matrix projective synchronization (EMPS) and asymptotic matrix projective synchronization (AMPS), respectively. It should be stressed that only the crucial node is controlled and only the state of the crucial node is applied to design the EMPS and AMPS controllers. This is promising to reduce the control cost as much as possible and to improve the feasibility of our MPS strategies. The effectiveness and feasibility of our MPS schemes are rigorously proved in theory as well as verified by two numerical examples.

Λ − φ generalized synchronization: application to fractional hyperchaotic systems with arbitrary dimensions and orders

This paper investigates the generalized synchronization between non-identical fractional-order systems characterized by different dimensions and different orders. The generalized synchronization combines the inverse matrix projective synchronization with the generalized synchronization. In particular, the proposed approach enables generalized synchronization to be achieved between n-dimensional master system and m-dimensional slave system in different dimensions. The technique, which exploits nonlinear controllers, stability property of integer-order linear systems and Lyapunov stability theory, proves to be effective in achieving the generalized synchronization. Finally, the approach is applied between 4-D and 5-D fractional hyperchaotic systems with the aim to illustrate the capabilities of the novel scheme proposed herein.

Combination Projective Synchronization in Fractional-Order Chaotic System with Disturbance and Uncertainty

In this manuscript, we study combination projective synchronization (CPS). In CPS, matrix projective combination synchronization (MPCS) and inverse matrix projective combination synchronization (IMPCS) between non-identical fractional-order complex chaotic systems subjected to uncertainty and external disturbance is investigated. Matrix projective synchronization (MPS) and inverse matrix projective synchronization is obtained when the scaling factor is a constant matrix, which gives the assurance of high security in secure communication and image encryption. Based on the Lyapunov stability theory and appropriate active control technique, the MPCS and IMPCS between two master systems and one slave system has been achieved. Based on the MPCS synchronization, a scheme of secure communication is presented, and the message signals are transmitted using the chaotic signal masking method. Finally, numerical simulations have been provided, which shows that our theoretical results are in complete agreement will the graphical one.

Global adaptive matrix-projective synchronization of delayed fractional-order competitive neural network with different time scales

In this paper, by using the adaptive control method, the global matrix-projective synchronization of delayed fractional-order competitive neural network with different time scales is researched for the first time. Firstly, the fractional-order global matrix-projective synchronization is defined. Then, in order to achieve the matrix-projective synchronization, the sufficient condition is obtained under an adaptive controller and its effectiveness is proved by combining the fractional-order Barbalat theory with a suitable Lyapunov–Krasovskii functional as well as some fractional-order differential inequalities. And all unknown parameters are identified and estimated to the fixed constants successfully. Finally, as applications, a numerical example with simulations is employed to demonstrate the feasibility and efficiency of the new synchronization analysis.

A New Matrix Projective Synchronization of Fractional-Order Discrete-Time Systems and its Application in Secure Communication

In this study, it is proposed that a new matrix projective synchronization of fractional-order (FO) chaotic maps in discrete-time. A new synchronization error is introduced and a control law is constructed, which makes the synchronization error converge towards zero in sufficient time under the stability theory of linearization method of FO systems. Numerical simulation results are presented to illustrate the feasibility of the scheme. Finally, a secure communication scheme based on FO discrete-time (FODT) systems was proposed.

Function matrix projective synchronization of non-dissipatively coupled heterogeneous systems with different-dimensional nodes

This paper regards function matrix projective synchronization of two different non-dissipatively coupled complex dynamical networks for different dimensions and different nodes. In this kind of complex dynamical networks the internal delays are different from the coupled delays. By using Lyapunov stability theory, using mathematical induction, two different hybrid feedback controllers are built to realize the function matrix projective synchronization. Compared with the existing results, the coupling matrices do not need to be symmetric or diffuse. By giving a numerical simulation we explain the validity and appropriateness of our conclusion.

Matrix projective cluster synchronization for arbitrarily coupled networks with different dimensional nodes via nonlinear control

SummaryRecently, many control mechanisms have been put forward to realize the cluster synchronization of networks. However, these results are derived by making very conservative assumptions on the network's nodes and its outer coupling configuration matrix, which greatly limit these results' potential applications. This paper focuses on weakening these restrictive assumptions as much as possible and further proposes nonlinear controllers for our general networks to achieve matrix projective cluster synchronization (MPCS). One outstanding characteristic of our introduced network model is that the state dimensions of the nodes in different clusters are different, which extends the case that the network is composed of identical dimensional nodes. Another distinctive characteristic is that no other restrictions are imposed on our network's outer coupling matrix except for being bounded, namely, both the outer coupling matrix and its block matrices may be time varying and nonzero‐row‐sum. Besides, the exponential MPCS (EMPCS) and asymptotic MPCS (AMPCS) are defined to make a better description of the cluster synchronization phenomena in this paper. It is worth noted that these two definitions extend the existing ones about cluster synchronization. Furthermore, by means of rigorous theoretical analysis, some nonlinear control strategies are proposed for our network to realize EMPCS or AMPCS. At last, a proper numerical example along with two control patterns is presented to demonstrate the validity and effectivity of our results in this paper.

A New Matrix Projective Synchronization and Its Application in Secure Communication

In this paper, a new matrix projective synchronization for chaotic (hyperchaotic) maps is proposed. The novel scheme called P-M synchronization is presented in this paper, since it combines two matrix projective synchronization schemes (one is based on the invertible matrix $P$ , and the other is based on matrix $M$ ). Compared to the regular matrix projection synchronization, the required matrix of the proposed scheme don’t change with different master systems. Under the framework of classical Lyapunov stability theory, a state feedback controller is selected to realize global synchronization. In addition, simulation results are reported, to highlight the capabilities of the P-M synchronization proposed in this paper. Finally, a speech secure communication scheme based on the P-M synchronization is implemented, implying the P-M synchronization can be applied into secure communication filed.

Secure Multiple-Input Multiple-Output Communications Based on F–M Synchronization of Fractional-Order Chaotic Systems with Non-Identical Dimensions and Orders

This paper investigates the F – M synchronization between non-identical fractional-order systems characterized by different dimensions and different orders. F – M synchronization combines the inverse generalized synchronization with the matrix projective synchronization. In particular, the proposed approach enables the F – M synchronization to be achieved between an n-dimensional master system and an m-dimensional slave system. The developed approach is applied to chaotic and hyperchaotic fractional systems with the aim of illustrating its applicability and suitability. A multiple-input multiple-output (MIMO) secure communication system is also developed by using the F – M synchronization and verified through computer simulations.

Dynamical analysis of a new fractional-order Rabinovich system and its fractional matrix projective synchronization

This paper constructs a new physical system, i.e., the fractional-order Rabinovich system, and investigates its stability, chaotic behaviors, chaotic control and matrix projective synchronization. Firstly, two Lemmas of the new system's stability at three equilibrium points are given and proved. Next, the largest Lyapunov exponent, the corresponding bifurcation diagram and the chaotic behaviors are studied. Then, the linear and nonlinear feedback controllers are designed to realize the system's local asymptotical stability and the global asymptotical stability, respectively. It's particularly significant that, the fractional matrix projective synchronization between two Rabinovich systems is achieved and two kinds of proofs are provided for Theorem 4.1. Especially, under certain degenerative conditions, the fractional matrix projective synchronization can be reduced to the complete synchronization, anti-synchronization, projective synchronization and modified projective synchronization of the fractional-order Rabinovich systems. Finally, all the theoretical analysis is verified by numerical simulation.

Fractional matrix and inverse matrix projective synchronization methods for synchronizing the disturbed fractional‐order hyperchaotic system

In this paper, for synchronizing two actual nonidentical fractional‐order hyperchaotic systems disturbed by model uncertainty and external disturbance, the fractional matrix and inverse matrix projective synchronization methods are presented and the methods' correctness and effectiveness are proved. Especially, under certain degenerative conditions, the methods are reduced to study the complete synchronization, antisynchronization, projective (or inverse projective) synchronization, modified (or modified inverse) projective synchronization, and stabilization problem for the disturbed (or undisturbed) fractional‐order hyperchaotic systems. In addition, as the fractional matrix and inverse matrix projective synchronization methods' applications, the fractional‐order hyperchaotic Chen and Rabinovich systems disturbed by model uncertainty and external disturbance are constructed, and the matrix and inverse matrix projective synchronizations between the two disturbed systems are achieved, respectively. This work constructs a basic theoretical framework of fractional matrix and inverse matrix projective synchronization methods and provides a general method for synchronizing the actual disturbed fractional‐order hyperchaotic systems that are related to science and engineering.

Matrix projective synchronization for time-varying disturbed networks with uncertain nonlinear structures and different dimensional nodes

Abstract This paper investigates the matrix projective synchronization (MPS) problem for time-varying disturbed complex dynamical networks (CDNs) with uncertain nonlinear structures and different dimensional nodes. In order to well describe the practical networks, the unavoidable unknown nonlinear structure and the inevitable uncertain disturbance of each node are considered into our network model, which is different from the previous works. Besides, it is worth pointing out that the outer coupling configuration matrix, which represents the coupling strength and the topological structure, is not restricted by the dissipatively coupled condition in this paper. The definition of MPS is also introduced for the networks with different dimensional nodes. Moreover, several MPS schemes are respectively put forward for our network model according to the norm bound of the uncertain nonlinear structures and disturbances being unknown or not. Finally, two proper examples associated with numerical simulations are given to verify the effectiveness of our theoretical results.